4,881 research outputs found
Representing convex geometries by almost-circles
Finite convex geometries are combinatorial structures. It follows from a
recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set
of planar convex polygons such that with respect to geometric
convex hulls is a locally convex geometry and every finite convex geometry can
be represented by restricting the structure of to a finite subset in a
natural way. An \emph{almost-circle of accuracy} is a
differentiable convex simple closed curve in the plane having an inscribed
circle of radius and a circumscribed circle of radius such that
the ratio is at least . % Motivated by Richter and
Rogers' result, we construct a set such that (1) contains
all points of the plane as degenerate singleton circles and all of its
non-singleton members are differentiable convex simple closed planar curves;
(2) with respect to the geometric convex hull operator is a locally
convex geometry; (3) as opposed to , is closed with respect
to non-degenerate affine transformations; and (4) for every (small) positive
and for every finite convex geometry, there are continuum
many pairwise affine-disjoint finite subsets of such that each
consists of almost-circles of accuracy and the convex geometry
in question is represented by restricting the convex hull operator to . The
affine-disjointness of and means that, in addition to , even is disjoint from for every
non-degenerate affine transformation .Comment: 18 pages, 6 figure
On the O(1) Solution of Multiple-Scattering Problems
In this paper, we present a multiple-scattering solver for nonconvex geometries such as those obtained as the union of a finite number of convex surfaces. For a prescribed error tolerance, this algorithm exhibits a fixed computational cost for arbitrarily high frequencies. At the core of the method is an extension of the method of stationary phase, together with the use of an ansatz for the unknown density in a combined-field boundary integral formulation
A 3-Manifold with no Real Projective Structure
We show that the connected sum of two copies of real projective 3-space does
not admit a real projective structure. This is the first known example of a
connected 3-manifold without a real projective structure.Comment: Minor corrections suggested by refere
- …